Rayleigh-Ritz Method

Basic Concept

The basic idea in the finite element method is to find the solution of a complicated problem
by replacing it by a simpler one. Since the actual problem is replaced by a simpler one
in finding the solution, we will be able to find only an approximate solution rather than
the exact solution. The existing mathematical tools will not be sufficient to find the exact
solution (and sometimes, even an approximate solution) of most of the practical problems.
Thus, in the absence of any other convenient method to find even the approximate solution
of a given problem, we have to prefer the finite element method. Moreover, in the finite
element method, it will often be possible to improve or refine the approximate solution by
spending more computational effort.

In the finite element method, the solution region is considered as built up of many
small, interconnected subregions called finite elements.Since it is very difficult to find the
exact response (like stresses and displacements) of the machine under any specified cutting
(loading) condition, this structure is approximated as composed of several pieces.In each piece
or element, a convenient approximate solution is assumed and the conditions of overall equilibrium of
the structure are derived. The satisfaction of these conditions will yield an approximate solution for
the displacements and stresses.